Of the non-experience molecular orbital method, the above-described MCSCF method and CI method have been used as typical computing methods that can use electron correlation. According to the MCSCF method, total energy E and the derivative thereof with respect to an atomic-nucleus coordinate, that is, the energy gradient (a force applied on the atomic nucleus) are given as below.
                    E        =                                            ∑              ab                              M                ⁢                                                                  ⁢                O                                      ⁢                                          γ                ab                            ⁢                              h                ab                                              +                                    1              2                        ⁢                                          ∑                abcd                                  M                  ⁢                                                                          ⁢                  O                                            ⁢                                                Γ                  abcd                                ⁡                                  (                                      ab                    ❘                                          c                      ⁢                                                                                          ⁢                      d                                                        )                                                                                        (        1        )                                                      ∂            E                                ∂            q                          =                                            ∑              ab                              M                ⁢                                                                  ⁢                O                                      ⁢                                          γ                ab                            ⁢                                                ∂                                      h                    ab                                                                    ∂                  q                                                              +                                    1              2                        ⁢                                          ∑                abcd                                  M                  ⁢                                                                          ⁢                  O                                            ⁢                                                Γ                  abcd                                ⁢                                                      ∂                                          (                                              ab                        ❘                                                  c                          ⁢                                                                                                          ⁢                          d                                                                    )                                                                            ∂                    q                                                                                                          (        2        )            Here, MO indicates a molecular orbital and q indicates any of the nuclei x, y, and z of an atom forming a molecule. γ and Γ indicate functions of a coefficient C of an electron configuration obtained through a solution that will be described later. hab and (ab|cd) are a 1-electron integration and a 2-electron integration at a molecular-orbital base and are obtained through hrs and (rs|tu) at an atomic orbital base. The hrs and (rs|tu) are defined as below.
                                          h            rs                    =                                    ∫                              -                ∞                            ∞                        ⁢                                          ∫                                  -                  ∞                                ∞                            ⁢                                                ∫                                      -                    ∞                                    ∞                                ⁢                                                                            x                      r                                        ⁡                                          (                                              r                        1                                            )                                                        ⁢                  h                  ⁢                                                                          ⁢                                                            x                      s                                        ⁡                                          (                                              r                        1                                            )                                                        ⁢                                      ⅆ                                          r                      1                                                                                                          ,                                  ⁢                  h          =                                    -                              ℏ                2                                      ⁢                                          ∇                2                            ⁢                              -                                                      ∑                                          i                      =                      1                                                              Nuc                      .                                                        ⁢                                                                                    Z                        i                                                                                                                                                  r                            1                                                    -                                                      R                            i                                                                                                                                        ⁢                                                                                  ⁢                                          (                                              where                        ⁢                                                                                                  ⁢                                                  (                                                      ℏ                            =                                                                                          h                                /                                2                                                            ⁢                              π                                                                                )                                                                    )                                                                                                                              (        3        )                                          (                      rs            ❘            tu                    )                =                              ∫                          -              ∞                        ∞                    ⁢                                    ∫                              -                ∞                            ∞                        ⁢                                          ∫                                  -                  ∞                                ∞                            ⁢                                                ∫                                      -                    ∞                                    ∞                                ⁢                                                      ∫                                          -                      ∞                                        ∞                                    ⁢                                                            ∫                                              -                        ∞                                            ∞                                        ⁢                                                                                            x                          r                                                ⁡                                                  (                                                      r                            1                                                    )                                                                    ⁢                                                                        x                          s                                                ⁡                                                  (                                                      r                            1                                                    )                                                                    ⁢                                              1                                                                                                                                      r                              1                                                        -                                                          r                              2                                                                                                                                                    ⁢                                                                        x                          t                                                ⁡                                                  (                                                      r                            2                                                    )                                                                    ⁢                                                                        x                          u                                                ⁡                                                  (                                                      r                            2                                                    )                                                                    ⁢                                              ⅆ                                                  r                          1                                                                    ⁢                                              ⅆ                                                  r                          2                                                                                                                                                                            (        4        )            Here, h indicates Planck's constant. Further, Nuc. stands for Nuclei and indicates the number of atoms. Zi indicates the electrical charges of the nuclei and Ri indicates the positions of the nuclei. The relationship between s molecular orbital φa and an atomic orbital χx is shown as below.
                              Φ          a                =                              ∑            r            N                    ⁢                                    C              ra                        ⁢                          X              r                                                          (        5        )            The integrations of the Equations (3) and (4) are transformed from the atomic orbital base to the molecular orbital base according to the relationship, as shown below.
                              h          ab                =                              ∑            r            N                    ⁢                                    ∑              s              N                        ⁢                                          c                ra                            ⁢                              c                sb                            ⁢                              h                rs                                                                        (        6        )                                          (                      ab            ❘                          c              ⁢                                                          ⁢              d                                )                =                              ∑            r            N                    ⁢                                    ∑              s              N                        ⁢                                          ∑                t                N                            ⁢                                                ∑                  u                  N                                ⁢                                                      c                    ra                                    ⁢                                      c                    sb                                    ⁢                                      c                    tc                                    ⁢                                                            c                      ud                                        ⁡                                          (                                              rs                        ❘                        tu                                            )                                                                                                                              (        7        )            Here, N indicates the number of atomic orbitals. Although the present invention is provided under the assumption that the number of the atomic orbitals is one thousand or more, this method can be used for a case where the number is less than that. ca is a transformation matrix shown in Equation (5) for transforming the atomic orbital into the molecular orbital and referred to as a molecular-orbital coefficient. csb is another molecular-orbital coefficient. According to the MCSCF method, both the electron-configuration coefficient C and the molecular-orbital coefficient c are obtained by the variational method. However, the CI method is different from the above-described method in that only the coefficient C is obtained.
The electron-configuration coefficient C is obtained through the following equations.
                                          ∑            J                          C              ⁢                                                          ⁢              S              ⁢                                                          ⁢              F                                ⁢                                    (                                                H                  IJ                                -                                                      δ                    IJ                                    ⁢                  E                                            )                        ⁢                          C              J                                      =        0                            (        8        )                                          H          IJ                =                                            ∑              ab                              M                ⁢                                                                  ⁢                O                                      ⁢                                          γ                ab                IJ                            ⁢                              h                ab                                              +                                    1              2                        ⁢                                          ∑                abcd                                  M                  ⁢                                                                          ⁢                  O                                            ⁢                                                Γ                  abcd                  IJ                                ⁡                                  (                                      ab                    ❘                                          c                      ⁢                                                                                          ⁢                      d                                                        )                                                                                        (        9        )            
Here, CSF is a configuration state function. A wave function is given as the linear combination of this asymmetric determinant CSF. δIJ is Kronecker delta. Where an expression I=J stands, the value of δIJ is one. At all other times, the value thereof is zero.
A predetermined amount is required for determining the molecular-orbital coefficient, as shown below.
                              y          ac                =                              ∑            b                    ⁢                                    ∑              d                        ⁢                                          ∑                x                            ⁢                                                ∑                  y                                ⁢                                                      {                                                                                            (                                                      ab                            ❘                            xy                                                    )                                                ⁢                                                  Γ                          cdxy                                                                    +                                              2                        ⁢                                                  (                                                      ax                            ❘                            by                                                    )                                                ⁢                                                  Γ                          cxdy                                                                                      }                                    ⁢                                      u                    bd                                                                                                          (        10        )            Here, Ubd is a matrix relating to linear transformation of the molecular orbital.
In either the MCSCF method or the CI method, generation of a 2-electron integration (ab|cd) at the molecular-orbital base constitutes most part of the computing cost. According to the complete active space SCF (CASSCF) method, which is a typical method of the MCSCF method, electron excitation is allowed only within a predetermined molecular-orbital range, so as to simplify the formula. Where the number of molecular orbitals in the active space is determined to be n and the number of atomic-orbital bases therein is determined to be N, usually, the relationship between n and N is shown as n<<N. According to a known computing scheme, all the 2-electron integrations at the atomic-orbital bases are stored in a main memory or an external storage medium such as a disk, and transformation shown in Equation (7) is performed. The transformation algorithm is shown in FIGS. 5 and 6. In the case of a simple eight-deep DO loop including indexes a, b, c, and d of the molecular orbital and indexes r, s, t, and u of the atomic orbital, n4N4 multiplication is required. However, according to the above-described algorithm, the same result as that of the eight-deep DO loop can be obtained by executing a five-deep DO loop four times, and the operand is shown as nN4+n2N3+n3N2+n4N. For example, where n=10 and N=1000, the computing speed increases by about one thousand times. However, according to this method, many computer resources are required for storing the 2-electron integrations at the atomic-orbital bases and intermediate data midway through transformation. Therefore, this method is not suitable for calculating large-sized molecules.
In recent years, computers have achieved high-speed operation by using parallel processors. Therefore, according to either the MCSCF method or the CI method, the size of a molecule to be calculated by parallel processing needs to be increased and the computing cost needs to be decreased. The advantages of a parallel computer are shown below.
1. A high-speed operation computer can be achieved at low cost by connecting many commodity processors.
2. Through the use of many local memories of the processors, the computer obtains a large main-memory area, as a whole.
The 2-electron integration at the atomic-orbital base and that at the molecular-orbital base are independent of each other. Therefore, where these integrations are parallelized based on this characteristic, the following problems arise. That is to say, since all the 2-electron integrations at N4 atomic-orbital bases are required for calculating one 2-electron integration at the molecular-orbital base,
1. distribution processing for distributing part of the 2-electron integrations at the atomic-orbital bases to the processors is performed, whereby all the N4 2-electron integrations have to be collected by each of the processors, which generates intercommunications between all the processors and become a bottleneck in communications, even though the integration-computing time can be reduced by parallelization.
2. Since all the integrations at the N4 atomic-orbital bases need to be calculated in each of the processors for reducing the bottleneck in communications, the reduction of computing time through the parallelization cannot be achieved.
Thus, the above-described problems are mutually contradictory to each other.
Further, the computing performed according to Equation (2) and the integration of the molecular-orbital base are required for obtaining the derivative of a 2-electron integration with respect to a nuclear coordinate. In this case, problems same as the above-described problems occur.